Table Of ContentPreliminary analysis of a recent experiment by
8 F. A. Bovino
0
0
2
n Emilio Santos
a
J Departamento de F´ısica. Universidad de Cantabria.
0
Santander. Spain
1
] February 2, 2008
h
p
-
t
n Abstract
a
u
An analysis is made of the results of a recent polarization correla-
q
[ tion experiment by Bovino (unpublished) where about 60,000 data
have been obtained. I assume that the state of the photon pairs
1
v produced in the source (a non-linear crystal) are in a (sightly) non-
2 maximally entangled state and the most relevant non-idealities of the
7
set-up are taken into account. A comparison is made between the
5
1 predictions of quantum mechanics and a simple family of local hidden
. variables models with the result that the former is violated by more
1
0 than 4 standard deviations whilst the data are compatible with the
8
family of local models.
0
:
v The possibility of making a true discrimination between local realism
i
X and quantum mechanics, via loophole-free tests of Bell´s inequality, seems
r extremely difficult or impossible as shown by the fa¨ılure of the efforts made
a
during more than 40 years after Bell´s work. Thus the standard practice has
been to test inequalities derived from local realism plus the assumption of
fair sampling. These inequalities have been violated in many experiments[1].
However fair sampling is an assumption theoretically absurd (because it ex-
cludes all sensible hidden variables theories[2]) and it has been claimed to be
empirically refuted[3]. Thus, in order to make real progress we should test
inequalites valid for restricted, but sensible, families of local hidden variables
(LHV) theories such as the ones derived in Ref.[4] and sumarized in Ref.[5].
In this light I will analyze the results of a recent experiment by Bovino[6].
1
In the Bovino experiment the detection efficiency of photon counters has
been 55% and the overall efficiency about 17%[6]. Thus, assuming losses of
order 15% in filters[7], I may estimate that there is a collection efficiency
(“effective fiber coupling coefficients”) about 36%. Bovino used two-channel
polarizers and he has recorded a huge amount of data which correspond
to 4 coincidence rates and 4 single rates for 45 angles of Alice´s polarizers
combined each with 45 angles of Bob´s polarizers.
In the ideal case quantum mechanics predicts for the coincidence rates,
R , as a function of the angle, φ, between Alice´s and Bob´s polarizers, a
12
cosinus curve of the form
Q
R (φ ) = hR i[1+V cos(2φ)], (1)
12 j 12
exp
The disagreement of the experimental data, R (φ ), with this prediction
12 j
may be measured by the quantity
Q exp
∆ = R (φ )−R (φ ) , (2)
12 j 12 j
r
Xh i
where hR i and V are chosen in eq.(1) so that ∆ is a minimum. There are
12
4 coincidence rates for every one of the 45 positions of Bob´s polarizer, so
that we may determine 180 quantities like (2). The values obtained range
between about 0.01 and 0.05 with statistical errors of order 0.01 in all cases.
In about 2/3 of cases the deviation surpasses 2σ (standard deviations) and
in about 1/3 it surpasses 4σ. So we might conclude that there is a significant
disagreement of the experimental data with the quantum prediction, eq.(1).
However it is most appropriate to attribute the disagreement to the non-
idealities of the experimental set-up than to a true violation of quantum
mechanics. Thus a more sophisticated analysis is needed. Obviously the
non-idealities should be also taken into account in the comparison of the
results with LHV models.
It is rather obvious that the experiment is not loophole-free (which would
require global detection efficiencies greater than about 80%) so that it does
not refute the whole family of LHV theories (a family which I have labeled
LHV0 for short[5]). Also apparently the results do not refute a simple family,
defined in Ref.[4], which I have labelled LHV1. In fact, this family predicts
that the quantity ∆, eq.(2), should be larger than about 0.001, which is
indeed the case. In contrast the family of models labelled LHV2, which
restricts LHV1 with the additional assumption of “fair sampling” applied to
2
the collection efficiency (“effective fiber coupling coefficients”) and the filters
but notto detectors, seems tobeviolated. It predicts that∆should belarger
than 0.04, a constraint not fulfilled in about half the cases (the document
Santos3.xls sent by Bovino reports that ∆ - labelled D(eta) there - is about
0.12 but my calculation gives a value about 1/3 that of Bovino). In any case
the violation might be attributed to the non-idealities of the experimental
set-up.
In summary the previous (rather poor) analysis of the experiment seems
to imply that it is compatible with the family LHV1 (and therefore LHV0)
but disagrees with both the family LHV2 and quantum mechanics. In order
to get more interesting information a better analysis of the data is required,
which is made in the following.
The non-idealities of the experimental set-up are quite important, as is
shownbythefactthatthe4singleratesareverydifferent,rangingfromabout
70000 to 110000 counts in the (unspecified) time window. Furthermore, for
a fixed position of Bob´s analyzer Alice´s single rates depend on the angle
of Alice´s analyzers, with a variation up to 7% between the maximum and
the minimum value. Similar variations exist for the single rates when we
consider different values of Bob´s analyzers with Alice´s analyzers fixed.
(No significant variation exists in the single counts of Alice (Bob) when Bob
(Alice) polarizer is rotated, as is expected by the “no-signalling principle”
which forbids sending information at a distance.) I shall make the quantum-
mechanical analysis of the experiment by studying the state produced in the
non-linear crystal and how this state evolves in the travel of photons until
the detectors, as follows.
1. I should assume that in the nonlinear crystal the photons are produced
always in pairs (no single-photon productions) and that the two photons in a
pair are in some pure, entangled, quantum-mechanical state. The entangle-
ment may not be maximal, although close to maximal. Thus I will assume
that the state is of the form
1
| ψ >= [| H > ⊗ | V > −(1+γ) | V > ⊗ | H > ], (3)
1 2 1 2
2
1+(1+γ)
q
where γ is a real number such that |γ| << 1.
2. Only a fraction µ (µ ) of the photons going to Alice (Bob) are col-
a b
lected, so that the state of the photons in the optical fibers becomes a sta-
tistical mixture of
3
a) the initial two-photon state (3) with weight µ µ ,
a b
b)asingle-photonstatehorizontallypolarizedforAlicewithweightµ (1−
a
µ )/(2+2γ +γ2) ,
b
c) a single-photon state vertically polarized for Alice with weight µ (1−
a
µ )(1+γ)2/(2+2γ +γ2)),
b
d) a single-photon state horizontally polarized for Bob with weight µ (1−
b
µ )(1+γ)2/(2+2γ +γ2)),
a
e) a single-photon state vertically polarized for Bob with weight µ (1 −
b
µ )/(2+2γ +γ2) ,
a
f) the vacuum state with weight 1−µ −µ +µ µ .
a b a b
3. The transmittances of the polarizationanalyzers of Alice for horizontal
or vertical polarization are such that, when a beam of linearly polarized
light with intensity I arrives at the polarizer, the transmitted and reflected
in
intensities are, respectively,
1
I (θ) = I (T −t )cos2θ +t = I [(T +t )+(T −t )cos2θ(]4,)
a+ in a+ a+ a+ in a+ a+ a+ a+
2
Ia−(θ) = Iin(cid:2)(Ta− −ta−)cos2θ +ta−(cid:3) = 1Iin[(Ta− +ta−)+(Ta− −ta−)cos2θ],
2
(cid:2) (cid:3)
and similar for Bob with the changes a → b. The sum T +t is close to unity
whilst 0 < t << 1.
4. The quantum efficiencies of the 4 detectors are different, say ζ and
a+
ζ for Alice and ζ and ζ for Bob.
a− b+ b−
In the ideal situation the quantum prediction for the 4 coincidence prob-
abilities of state (3) would be[3]
1
2
P = [(1+γ)sinαcosβ −cosαsinβ] ,
++ 2
1+(1+γ)
1
2
P+− = 2 [(1+γ)sinαsinβ +cosαcosβ] ,
1+(1+γ)
1
2
P−+ = 2 [(1+γ)cosαcosβ +sinαsinβ] ,
1+(1+γ)
1
2
P−− = [(1+γ)cosαsinβ −sinαcosβ] , (5)
2
1+(1+γ)
where α (β) is the angle of Alice´s (Bob´s) polarization analyzer. Now
I introduce the most relevant non-idealities as follows. Due to losses and
4
absorptions, as explained above, there is a global factor
µ (T +t )ζ µ (T +t )ζ
a a+ a+ a+ b b+ b+ b+
in front of P++ and similarly for P+−,P−+ and P−−. In addition every cos2α
or sin2α should be preceded by either a factor (T − t )/(T + t ) or
a+ a+ a+ a+
a factor (Ta− −ta−)/(Ta− +ta−) (see eqs.(4)) and similarly ,with a → b, for
cosβ or sinβ. Thus I get
1
′ ′′
P = η η [1−V cos2φ+γ (cos2β −cos2α)+γ sin2αsin2β],
++ 4 a+ b+ ++
1
′ ′′
P+− = 4ηa+ηb−[1+V+−cos2φ−γ (cos2β +cos2α)−γ sin2αsin2β],
1
′ ′′
P−+ = 4ηa−ηb+[1+V−+cos2φ+γ (cos2β +cos2α)−γ sin2αsin2β],
1
′ ′′
P−− = 4ηa−ηb−[1−V−−cos2φ−γ (cos2β −cos2α)+γ sin2αsin2β(]6,)
where
2γ +γ2 γ2
′ ′′
φ ≡ α−β, γ ≡ , γ ≡ , η ≡ µ T ζ ,
2+2γ +γ2 2+2γ +γ2 a+ a a+ a+
and similarly for the remaining parameters η. The term V is given by
++
(T −t )(T −t )
a+ a+ b+ b+
V =
++
(T +t )/(T +t )
a+ a+ b+ b+
and similarly for V+−,V−+ and V−− (see eqs.(4)) . The correction for finite
transmittance (i. e. the fact that t > 0) has not been taken into account
in the terms containing the parameter γ. Indeed it should be realized that
γ ∼ 0.1 and t ∼ 0.01, therefore I am neglecting terms of order 0.001 with
respect to the main term, whilst I do not neglect terms of order γ2 ∼ 0.01.
AlsowemaycalculatetheprobabilitiesforsinglecountsbyAlice,Pa+,Pa−,
and Bob, Pb+,Pb− , respectively, getting
1 1
′ ′
Pa+ = 2ηa+[1−γ cos2α], Pa− = 2ηa+[1+γ cos2α],
1 1
′ ′
Pb+ = 2ηb+[1+γ cos2β], Pb− = 2ηb+[1−γ cos2β]. (7)
The number of counts within one time window should be obtained by multi-
plying the probabilities (6) or (7) times R , this being the number of photon
0
pairs produced in the source within the window. That is R = R P , etc.
++ 0 ++
5
Thequestion iswhether allthedataofBovino´sexperiment maybefitted
to eqs.(6) and (7) with 10 free parameters, namely η , η , η , η , γ, V ,
a+ b+ a− b− ++
V+−,V−+,V−− and R0. The analysis simplifies a lot as follows. From eqs.(7) I
get the global efficiencies by averaging over angles, that is
ηa+ = 2hPa+i,ηb− = 2hPb−i,ηa− = 2hPa−i,ηb+ = 2hPb+i. (8)
Now I may eliminate the efficiencies η , etc. in eqs.(6) using eqs.(8) and
a+
pass from probabilities to count numbers by multiplication times R in the
0
appropriate places. Thus I get
R R
0 ++ ′ ′′
= 1−V cos2φ+γ (cos2β −cos2α)+γ sin2αsin2β,
++
hR ihR i
a+ b+
R0R+− ′ ′′
= 1+V+−cos2φ−γ (cos2β +cos2α)−γ sin2αsin2β,
hRa+ihRb−i
R0R−+ ′ ′′
= 1+V−+cos2φ+γ (cos2β +cos2α)−γ sin2αsin2β,
hRa−ihRb+i
R0R−− ′ ′′
= 1−V−−cos2φ−γ (cos2β −cos2α)+γ sin2αsin2β(.9)
hRa−ihRb−i
Hence I may define the following average coincidence detection probability,
P (α,β),
P ≡ 16R f (10)
0
f ≡ R+−(α,β) + R−+(α,β) + R++ α+ π4,β − π4 + R−− α+ π4,β − π4 .
hRa+ihRb−i hRa−ihRb+i hR(cid:0) a+ihRb+i (cid:1) hR(cid:0)a−ihRb−i (cid:1)
The quantum prediction for P depends on the angles α and β only via the
combination α−β ≡ φ and it is rather simple, namely
1
PQ(φ) = [1+V cos2φ], P (φ) ≡ 16R f, (11)
0
4
where
1 2γ2
V ≡ 4(V++ +V+− +V−+ +V−− − 2+2γ +γ2).
(It may be realized that the same result is obtained putting α− π,β + π
4 4
as the argument of R++ and R−−.) At this moment I stress that, for the dis-
(cid:0) (cid:1)
crimination between quantum mechanics and LHV models, the combination
of rates eq.(10) is more appropriate than the fashionable combination
R++(φ)+R−−(φ)−R+−(φ)−R−+(φ)
U(φ) = , (12)
R++(φ)+R−−(φ)+R+−(φ)+R−+(φ)
6
forwhichquantummechanicspredictsasimpleexpression, namelyV cos(2φ),
only in the ideal case.
A good fit of the data into eqs.(10) and (11), with R and V as free
0
parameters, seems to be a necessary condition for the compatibility of the
experiment withquantummechanics. However, even ifagoodfitisnotpossi-
ble, stilltheexperiment maybecompatiblewithquantum mechanics because
the state produced in thesource may be different fromeq.(3)and/or there are
additional non-idealities not included in the previous analysis. Consequently
proving an empirical violation of quantum mechanics is extremely difficult or
impossible in actual experiments. Similarly refuting local realism is impos-
sible whenever the global detection efficiency does not surpasse about 80%,
a well known fact.
Nevertheless, even if the Bovino experiment[6] does not allow a rigorous
discrimination between quantum mechanics and local realism, interesting
information may be obtained by studying the agreement, or disagreement,
of the data with some simple LHV models departing but slightly from the
quantum predictions eqs.(6) or (7). This study would require to define a
simple family of LHV models and to find whether the data agree with either
the quantum predictions or the said simple LHV models. Constructing LHV
models appropriate for a comparison with the quantum eqs.(6) or (7) is a
most interesting aim for the near future, but from my experience I may guess
that the results will be as follows. The model predictions will be of the form
given by eq.(37) of [4], that is (compare with eqs.(11))
1
PLHV (φ) = [1+V cos(2φ)+δ(φ)], (13)
4
δ(φ) being
8ε3 sin2(πη/2) 2 2
δ(φ) = 2 cos(2φ)−1+ η + |φ|−1 ,
3π (πη/2)2 η2 π
(cid:20) (cid:18) (cid:19)+(cid:21)
whereφ ∈ [−π/2,π/2]and() meansputting0ifthequantityinsidebrackets
+
is negative. The parameter η is an averaged detection efficiency and ε is the
solution of the equation
2
π −2ε+sin(2ε)cos(2ε) (πη/2)
= V . (14)
cos(2ε)[π −2ε+tan(2ε)] sin2(πη/2)
if ε > 0 or ε = 0 if the solution is negative.
7
Eqs.(10) and (13) may be rewritten taking account of the first two terms
of δ(φ) by means of small changes in the parameters R and V, that is
0
1 32ε3 π πη
PLHV (φ) ≡ 16R′f = 1−V′cos(2φ)+ |φ|− + . (15)
0 4 3π2η2 2 2
+
(cid:20) (cid:21)
(cid:16) (cid:17)
After that, the discrimination between quantum and LHV predictions will
consist of checking whether the quantum eq.(11) (with R0 and V as free pa-
′ ′
rameters) or the LHV eq.(15) (with R ,V and η as free parameters) may
0
be fitted to the data of the experiment. In practice we should make chi-square
fits of the experimental data into eq.(15) with 3 free parameters, namely
′ ′ ′
R ,V and η. The parameter ε is related to η and V by eq.(14). If η << 1
0
eq.(14) may be solved to order ε2 giving[4]
1 sin2(πη/2)2 1 sin2(πη/2)2
ε ≃ 1− ≃ V′ − , (16)
s2 V′(πη/2)2 s2 (πη/2)2
(cid:18) (cid:19)+ (cid:18) (cid:19)+
′
where the second equality is valid for V close to unity, as is usually the case.
I guess that fairly good fits exist for some set of values of η. If a good fit
is possible for η = 0 (in this case the LHV eq.(15) become identical to the
quantum eq.(11)) then the experiment is compatible with standard quantum
mechanics (“standard” means accepting the analysis leading to eqs.(6) and
(7)). If good fits are possible for η ≥ 0.17 then the experiment is compatible
with the family of local models defined in [4], a family labelled LHV1 in
[5]. If there are good fits for η ≥ 0.55 then the experiment would be also
compatible with the family defined as LHV2 in [5], which is more restrictive
than LHV1. However I do not think the latter will be the case in view of the
results of the rough analysis made at the beginning of this paper.
The fit of the data into the equations should be made for every one of the
46 sets of data corresponding to one Bob´s polarizer position each, rather
than a fit of the whole set of data. The reason is that in Bell tests it is
essential that the measured rates correspond to the same production rate
in the source. I suppose that in the Bovino experiment it is much easier
to guarantee the constancy of the production rate for every Bob´s polarizer
position than for the whole set of data.
In order to make a preliminary analysis of the experiment I have uses a
few data of “fiBob” = 90o. A simple consequence of the (LHV) eq.(13) is,
assuming η ≤ 1/2,
f (0)+f (π/2)−2f (π/4) 16ε3/(3πη) 4ε3
ν ≡ = ≃ , (17)
f (0)+f (π/2)+2f (π/4) 4+16ε3/(3πη) 3πη
8
′
whilst standard quantum mechanics predicts ν = 0. The value of V may be
obtained from
f (0)−f (π/2)
′
V = ,
f (0)+f (π/2)
and I get
V′ = 0.9760, νdata ≈ 0.00149±0.00032. (18)
(In the experiment no data for the angle π/4 have been reported and I have
used averages between the angles 44o and 46o.) In eq.(18) the statistical
error has been calculated from the data for φ = π/4, the errors in other data
making a negligible change. The result eq.(18) suggests that the experiment
disagrees with standard quantum predictions (ν = 0) by about 4.5 standard
deviations. In order to study the agreement with the family of local models
LHV1 we should search for the value of η which gives νLHV ≃ 0.015. We
realize that this is achieved by η = 0.225, which gives ε = 0.092 using
eq.(16), whence eq.(17) leads to agreement sith the empirical result eq.(18).
Consequently the data are compatible with the family LHV1 but refute the
family LHV2 because η = 0.225 is greater than the overall efficiency 0.17
but smaller than the quantum efficiency of detectors 0.55.
As another example, a similar analysis of data of “fiBob” = 40o gives
V′ = 0.9521, νdata ≈ 0.00142±0.00047, (19)
that is a result departing from standard quantum predictions, ν = 0, by 3
standard deviations but in agreement with the family LHV1 of local models.
In summary, although a more complete analysis is needed, the data of the
Bovino experiment seem to depart from standard quantum predictions by an
amount which agrees in sign and order of magnitude with the predictions of
a simple family of LHV models[4]. However I should not conclude that a real
violation of quantum mechanics has taken place. In fact, there may be non-
idealities of the set-up not taken into account in the quantum-mechanical
analysis leading to eqs.(6) or (7). Nevertheless if a more complete analysis
confirms that the data agree with the predictions of simple LHV models,
eq.(15), better than with the standard quantum prediction, eq.(11), that
would reinforce my conjecture that non-idealities of experimental set-
ups tend to save local realism. If the non-idealities may be explained
within quantum mechanics, then the conjecture would be that quantum
mechanics and local realism are compatible at the empirical level,
Bell´stheorembeingtrueonlyforanidealized(incorrect)versionofquantum
mechanics.
9
References
[1] M. Genovese, Phys. Rep. 413, 319 (2005).
[2] E. Santos, Found. Phys. 34, 1643 (2004); c-print
arXiv:quant-ph/0410193.
[3] G. Adenier and A. Yu. Khrennikov, J. Phys. B: At. Mol. Opt. Phys. 40,
131 (2007).
[4] E. Santos, Eur. Phys. Jour. D 42, 501 (2007) ; c-print
arXiv:quant-ph/0612212.
[5] E. Santos, Eur. Phys. Jour. D (2007). In press.
[6] F. A. Bovino (2007), unpublished. Private communication.
[7] F. A. Bovino et al., c-print arXiv:quant-ph/0303126.
10
